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It is known that every number can be represented by some red-blue Hackenbush stalk (see here, for instance). What values can be represented by red-blue-green Hackenbush stalks? In addition, what games (i.e. not in canonical form) can be represented by Hackenbush stalks? (For example, we can represent sums of nimbers easily by just taking their XOR, but it is impossible to represent the non-canoical form of sums of nimbers with a single stalk.)

Similarly, is it possible to answer the same question for Hackenbush "trees", i.e. graphs that have only one edge adjacent to the ground?

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    $\begingroup$ Assuming finite stalks, the moves that survive in canonical form are all green moves (for both sides) and the highest blue (respectively red) edge if and only if there is no green edge further up the stalk from it. I'd be surprised if there were a nice description of general stalks in terms of common elementary game values. $\endgroup$
    – Gabriel C. Drummond-Cole
    Commented Dec 27, 2020 at 8:54
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    $\begingroup$ The following question (concerning forests, not just stalks) is at least somewhat relevant: mathoverflow.net/q/267112/17064 $\endgroup$
    – Gro-Tsen
    Commented Dec 27, 2020 at 13:40
  • $\begingroup$ Thanks for your replies! Is there a known way to evaluate red-blue-green stalks, like there is for red-blue stalks and green stalks? $\endgroup$
    – flame
    Commented Dec 27, 2020 at 17:43
  • $\begingroup$ @flame What would you want the output of such an evaluation to look like? $\endgroup$
    – Gabriel C. Drummond-Cole
    Commented Dec 27, 2020 at 18:22
  • $\begingroup$ @GabrielC.Drummond-Cole Similar to an algorithm for evaluating red-blue Hackenbush positions. Though I suppose one doesn't exist yet (and it may be impossible to come up with a nice algorithm in general), according to my research. $\endgroup$
    – flame
    Commented Dec 27, 2020 at 19:37

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